How to read mathematics

How to Read Mathematics?

Mathematics is a language that can neither be read nor understood without initiation.

You can’t read a math book the way you read other books. It takes a special approach to read maths. Mathematics is not like a novel reading. It can never be understood if you go through it starting page 1 to the last page in one breath. In order to understand mathematics you must develop a reading protocol to get benefit. Poetry calls for a different set of strategies than fiction, and fiction requires another set of strategies than nonfiction. Mathematics has a reading protocol all its own, and just as we learn to read literature, we should learn to read mathematics.  Always adopt pen and paper for doing mathematics otherwise you are simply deceiving yourself. You will never be able to understand mathematics unless or until you have done it in writing.

Reading mathematics is difficult because mathematics is difficult. It is considered the fiercest subject in the world and this is due to our pre notion approach. Mathematics is indeed the success gate of science and that’s why the great mathematician C F Gauss said-- Mathematics is the queen of all subjects.

Mathematics has the distinction of having the dense writing style in the text book which is not easy to understand. It is not true. Mathematics text requires more involvement from the reader than most texts in other subject. Hence, reading mathematics takes longer time and attempting to read mathematics too fast results to frustration. According to Adler- if a text is worth reading at all it is worth three readings at least. Mathematics is not a novel where you become absorbed in the plots and characters. The scene depicted is exaggerated but mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. Both a mathematics article and a novel are telling a story and developing complex ideas, but a math article does the job with a tiny fraction of the words and symbols of those used in a novel. Let us learn the technique to read mathematics.

1. Synthetic Reading of the Book: - In the synthetic reading, the reader proceed from the parts to the whole. The parts are the building blocks; you can’t jump on the second until you finish the first. Learn the basics and make a strong base. Reading mathematics is not at all a linear experience. Understanding the text requires cross references, scanning, pausing and revisiting. You can get a clear picture of next chapter if you have prepared and done well in the previous chapter. Nobody understands something complex on first reading. If after devoting for an hour you understand only one or two main points of the chapter don’t be sad about this. It doesn’t mean that there is anything wrong with you. The first reading just lays the framework for you to fill in later with details.

            A) Read the chapter introduction and each section summary.                                             B) Skim the chapter; circle the new words that you don’t understand. Consult the      dictionary for those words. Once you understand the concept erase the circle.                 For better clarification, if need be, consult your teacher.                          

C) Read with concentration. While reading the textbook, highlight the important result, formula, theorem etc.                                                                                    D) Before moving to the main topic, go through the example with noticing each and every step properly. In the example you might find many steps missing. Don’t jump to the conclusion that you will do the same in the examination. Try to find the missing steps with proper reasoning. Write those steps in text book for better understanding. Remember you learn math by doing, not by reading. If you don’t understand anything, refer to another math textbook, computer software program or consult your instructor, teacher for better understanding.                                  E) Once you find yourself equipped with the content of chapter, explain it to your friend.  If there is no one else available, can you explain it aloud, without stumbling? If you can do that, you probably understand it.

1. Learn Mathematical Symbols and vocabulary: - Before you attempt to understand the mathematical terms, you should learn the mathematical symbols commonly used in your textbook. In Algebra, you may see symbols like Σ (sigma) for addition, Π for multiplication. In trigonometry you may see the symbols like α (Alpha), β (Beta), γ (Gamma), δ (Delta), θ (Theta), φ (Phi) etc. Make a list of symbol and learn it. If possible, copy the list of math symbol from your math book or math dictionary and paste it on the wall so that you can revise the list every now and then. Moreover, it is also advisable to consult math dictionary for new words. Mathematics obtains much of its power by constructing a very precise vocabulary. A strong vocabulary can help you to understand the content of the book more easily and effectively. The most fundamental issue involved in reading mathematics for meaning is to get some sense of the mathematical words, phrases and applications. Mathematical text is a combination of jargon. Therefore we should not attempt to read mathematics as we would other type of reading. Reading mathematically is more than reading the printed words on the page. It requires linking the words with the mathematical ideas that are involved. So , making sense of mathematical prose is a complex process involving understanding mathematical terms and this can be done only when you learn the math vocabulary.
2. Learn Mathematical statement, formula: - In order to understand the mathematical concepts, you must learn the formula, statement of theorems, axioms etc. so that you may have the better understanding of the chapter. Try to learn the way how the formulas are generalized. It is better to learn the way formula is constructed rather than remembering the bags of formula. Make a poster of formula and paste it on the wall so that you can have a look of it every morning. Spend at least 5-10 minutes daily in revising these. You can develop your own way of remembering the formula. It is advisable to consult your teacher and ask him/ her to teach you the innovative ways of making and learning formula. You can also enquire about Pascal Triangle that is very helpful in understanding the binomial expansion of any positive power. Likewise you can also learn AFTER SCHOOL TO COLLEGE to remember the mathematical formula for sin(90+ө) , sin (180 ± ө), sin (270 ±ө) and sin (360±ө) etc. I still remember my first day in Trigonometry class when teacher wrote PANDIT BADRI PRASAD, HARI HARI BOLE to define the different Trigonometric ratios such as sine, cos, tan etc. I shall therefore put much emphasis on learning different techniques to learn formula rather than just learning the hundreds of formula.
3. Don’t be a passive Reader: - Mathematics is all about putting an extra effort. Go the extra mile and do some research works to find out what the particular page you are struck into want to tell you. Always keep a pen and paper with you while you read math text book. Write steps and solve the step by your own rather than seeing what is written in the book. Many results must be given in your book of which the details are suppressed. It is expected from the readers to fill the gap. If you are doing the work without pen and paper, you may probably not be able to understand certain thing. Always remember—to a great extent, people think mathematically through writing. It is hard to do in your head. Read the chapter introduction and each section summary. Skim the reading material that will allow you to see that if problems presented in one chapter is being explained in next chapter or not. As you skim the chapter, circle the new words that you don’t understand and ask these words to your parents first and then to teacher in the next day if you don’t understand these new words after reading the assignment .Mathematics says a lot with a little. Math uses special words to mean specific things. Sometimes, words are used differently in math than in regular language. For example PRIME, SET, VOLUME, COMPLEX, REAL, RATIONAL etc have different meaning in maths than they usually do. Understanding maths term will help you understand topic. Remember a math text book is very difficult. It might take you half an hour to read and understand just one page so don’t get impatient in such situation. You must be an active partner. At every stage you must decide whether or not the idea being presented is clear. Ask yourself these questions: ---                                                                                                                          Why is this idea true?                                                                                                          Is there any better option to challenge the idea?                                                              Could I convince someone else that it is true?                                                                       Why did the author use a different argument?                                                                         Why this particular formula or theorem was used here?
4. Study the examples and figures: - Most textbooks include examples with detailed solutions. The author of textbook design the exercise on the basis of examples discussed. Paying attention to the examples discussed provide an excellent opportunity for you to assess your readiness to begin the assigned exercises at the end of the section. If you go through the introduction of the chapter and read the example and correlate the matter, you may find the exercise handling is a cake walk for you.
5. Take a break when you are struck on a problem: - Many a time when you are trying to solve a math problem, you may force yourself to keep working at it until you find the solution. This may not be as useful as you think. If you get struck on one example, put it aside for a while.                                                                                                                                Take a break of 5-10 minutes.                                                                                                 Drink a glass of water or gat a snack.                                                                                          Do something completely different that will get your mind on something else. When you feel refreshed, go to the problem and read it with a fresh eye. This time you will have the solution. If problem still persists, ask your parents for immediate help.
6. Don’t make too much selection: - I have seen students making selection in reading. In the age of technology, students are aware of the pattern of the questions, blue prints of the syllabus and they make choices. A mediocre student selects the list of chapters that are easy to understand and estimate the necessary marks to pass by reading those few selected chapters. There is no harm in making selection of chapters but never thing that your ultimate goal is to pass in the examination with minimum passing marks. Go through all the chapters thoroughly and do all types of questions. Don’t make selection of important question at the very beginning. Remember                                                                                                                  Better know everything of something                                                                                    rather than something of everything                                                                                                       AND                                                                              Nothing like if you know everything of everything.                                              Selection of questions and chapters at the beginning will limited your knowledge and though you may feel good that this is the smartest way to pass the examination but it will do much harm to you in later phase of your life.
7. Put a special mark in your book: - Your teacher tells you to do some questions 5-10 times because that question is important. Your teacher with his experience knows those particular questions are important for the examination and lets you know in advance. My advice is in order to simplify the reading and also to select the important questions for your revision work before examination you make a regular habit of marking such questions with red ink to identify it and ease your learning during examination. You may mark with the initial letter of your name. Moreover, you can give special mark to those questions which you feel are difficult or require some special attention. You may write the special character at the first page of your book such as—R = Revision, I = Important, R*** = Revise it three  times etc. don’t include in this list those question which are based on the direct application of formula and which you think you can do.
8. Be a regular reader: - If you love watching cartoons/ serials on the Television and you miss a episode or two, you will still be in a position to judge what had happened in the last episode. But the same is not true for mathematics. Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. A mathematics article and a serial/ cartoon shown on TV are telling a story but a math article does the job with a tiny fraction of the words and symbols of those used in a serial/ cartoons. The TV serial uses language to evoke emotions and present themes which defy precise definition but the beauty in a mathematical article is in the elegant efficient way it concisely describe precise ideas of great complexity.                                                                         So if you are jumping one chapter or a single step in mathematics than there is greater chances that you fail to understand the whole concept and deflect the way. Always remember, mathematics can’t be mastered in a single day. You have to read mathematics regularly for at least 1 hour a day if you can’t put maximum in a particular day. Keeping yourself away for a week or two from math book will bring you to the mark zero and you have to pad up again to regain the knowledge because it is sure that you have forgotten almost 90% of what you have learnt a week or 15 days ago.  As you watch the television serial whether it is an entertainment channel or informative one, you must have seen that before the serial begins it recast the glimpses of the previous day. In the same manner, you must have to revise the odd words, formulas, important facts of the previous chapter before switching to the next.
9. Get your problem solved instantly: - Mathematics can be learnt if you resolve the problem instantly. Keep the phone number of your class teacher and your tutor and call him or her when you feel helpless in understanding a particular step or problem. It is better to consult the doctor when you feel uneasy rather than wait for the problem to grow. You can’t wait for a week to consult your class teacher to resolve the problem you face a week ago. Suppose you are reading a chapter from Algebra and while solving a particular sum you faced a problem. My simple advice is you first consult your parents. If they are good at mathematics then you will get the answer instantly. If you can’t get the answer, look at the example given in that chapter. If you have kept some reference book than you can consult the book too. If your labour fails in vain then don’t get hesitate in calling your teacher or tutor after all they are to help you. But don’t call them late night.
10. Keep the Reference Book/ Journal with you: - Reading mathematics is not at all a linear experience. Understanding the text requires cross references, scanning, pausing and revisiting. When you read a math book, you may or may not encounter several problems. Don’t assume that understanding each phrase, word, symbol etc. will enable you to understand the whole idea. This is like trying to see a portrait painting by staring at each square inch of it from the distance of your nose. You will see the detail, texture and colour but miss the portrait completely. A math article tells a story. Try to see what the story is before you delve into the details. You may not get the detail in your text book itself so you have to consult some other book or journals. A class text book has a limited number of problems and you may not get varieties of problem in it but a good reference book will help you to solve varieties of problem boosting your confidence. A good reference book will help you to understand the definition, properties of numbers, geometrical shapes etc and also the algorithm to solve problem of particular type.
11. Pay attention to your anxious feelings:- Some people feel like they are simply not able to learn math. They may have been unsuccessful in learning math earlier or may have been told that they could not do math. This is called math anxiety. Math anxiety has nothing to do with abilities. If you feel that you can’t do math simply doesn’t mean that you are unable to do math. The feeling can get in the way. If you see a problem that is difficult for you, you may unknowingly tell yourself that you can’t do it. A key to getting over math anxiety is to figure out what is going on and manage them until the problem over power you. Make a journal and write your good/ bad feeling in it. If you feel happy after solving a particular problem, write it down. In the similar way if you feel distressed or nervous when you fail to solve a problem, also write it. As you know thinking in math is related to doing. This exercise of writing your feeling allows your mind to critically analyse the problem. You analyse to anticipate problem areas when you write down your feeling and this way you are helping yourself in another way.
12. Don’t read too fast: - In the very beginning, I have told you that mathematics book should not be read as novel. You may read 10-20 pages in half hour if you are a avid novel reader but the same may not be true for mathematics. The same amount of hour in a math book may finishes with 1-2 pages depending on the chapter you are reading and how experienced you are at reading mathematics. There is no substitute of work and time. You can speed up your math reading skill by practicing. In novel reading, you may skip the unwanted paragraph but still you can understand what novel is trying to say. This may not be true in case you are reading math. A single paragraph may have hundreds of hidden facts that will be essential to understand the next paragraph. So, please be patience and hold your breath. Mathematics is called the queen of all subjects and you can’t make please queen in your zig-zag style of reading.
13. Practice, practice and practice: -       Mathematics is not a subject you learn in a single attempt. Keep pen and paper and do as many problems as is required to ensure that you understand the concept. The amount of practice may vary from person to person but you can’t skip practicing. You will want to practice a concept until it makes sense and until you are fluent at finding solutions to various problems within the concept readily. When you complete a set of questions in a row, you are probably to the point of understanding. Re- visit the same problem after a month to check whether you are still capable of doing the same problem with the same amount of easiness or not. Think of math the way one thinks about a musical instrument. Most of us don’t just sit down and play an instrument. We first take lessons, practice it several times before moving to the next lesson. A good musician takes out time to review and never stops practicing. Mathematics is like the same. You need to practice more and more. Do extra exercise. Go beyond what is asked for. If you are asked to do 20 odd questions, do it but never put yourself in a particular boundary. Buy another book and practice more and more until you reach to the point of fluency with the concept. Doing the extra practice questions will help you to grasp the concept more readily. Be sure to re- visit the exercise a few months later to ensure that you still have a grasp of it.
14. Discuss what you learnt with your friends:-  The best way to get good at math is to discuss what you have learnt to your friend. It is said- Two heads are better than one. When you discuss the problem in a group, you are clarifying the concept for you by looking at it in a different way. When you learn something new from your parents, tutor or book you read that is completely different from what you have learnt in class, never forget to share it with your friend. Remember what William Glasser says-                               10% of what we READ, 20% of what we HEAR, 30% of what we SEE , 50% of what we SEE and HEAR,70% of what is DISCUSSED with OTHERS ,80% of what is EXPERIENCED PERSONALLY , 95% of what we TEACH TO SOMEONE ELSE.                       
15. Read backwards and forwards: - Mathematical knowledge can’t be gained by straight reading. You have to move in all the directions. You may not fully learn something in chapter 1 until you are halfway through chapter 2 or chapter 3. Hence it is a good idea to look back once in a while over previous sections so that you can have the control of what you have learnt previously and your knowledge don’t get outdated.  

I hope my advice will help you to read mathematics in a planned manner and you know when any work is done with a full proof planning an outstanding result is sure to come.    

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Rajesh kumar Thakur

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Mathematics and Hindu Religion lecture series

Fermat

Pierre De Fermat

Born: - August 17, 1601                     Died: - January 12, 1665

Fermat was one of the leading mathematicians of early 17^th century. Pierre Fermat's father was a wealthy leather merchant and second consul of Beaumont- de- Lomagne. Although there is little evidence concerning his school education it must have been at the local Franciscan monastery. He was an amateur mathematician.

 He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius Plane loci to one of the mathematicians there. From Bordeaux Fermat went to Orléans where he studied law at the University and spent his working life as a magistrate in the small provincial town of Castres.

After he moved to Toulouse , he gained a new mathematical friend Carcavi.  In 1636 Carcavi went to Paris as royal librarian and made contact with Mersenne and his group. Mersenne's interest was aroused by Carcavi's descriptions of Fermat's discoveries on falling bodies, and he wrote to Fermat. Fermat replied on 26 April 1636 and, in addition to telling Mersenne about errors which he believed that Galileo had made in his description of free fall. Fermat had little interest in physical applications of mathematics. Even with his results on free fall he was much more interested in proving geometrical theorems than in their relation to the real world. Fermat sent a letter to Mersenne containing two problems on maxima which Fermat asked Mersenne to pass on to the Paris mathematicians and this was to be the typical style of Fermat's letters, he would challenge others to find results which he had already obtained. Roberval and Mersenne found that Fermat's problems in this first, and subsequent, letters were extremely difficult and usually not soluble using current techniques. They asked him to divulge his methods and Fermat sent Method for determining Maxima and Minima and Tangents to Curved Lines.

His reputation as one of the leading mathematicians in the world came quickly but attempts to get his work published failed mainly because Fermat never really wanted to put his work into a polished form. However some of his methods were published in Cursus mathematicus a work by Herigone. With Pascal, Fermat stands as one of the founder of mathematical theory of probability. Pierre de Fermat independently founded the new branch of mathematics called Analytical Geometry. This work led to violent controversies over question of priority with Rene Descartes. Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La géométrie. This manuscript was published posthumously in 1679 in "Varia opera mathematica", as Ad Locos Planos et Solidos Isagoge, ("Introduction to Plane and Solid Loci").^[

In Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation.

He is probably best known for his work on number theory. He also left one of the famous unsolved problems in maths called- Fermat’s last theorem. This theorem states that xⁿ + yⁿ = zⁿ has no non-zero integer solutions for x, y and z when n > 2. in the margin of Bachet's translation of Diophantus's Arithmetica , Fermat worte “ I have discovered a truly remarkable proof which this margin is too small to contain.” These marginal notes only became known after Fermat's son Samuel published an edition of Bachet's translation of Diophantus's Arithmetica with his father's notes in 1670. Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries. The truth of Fermat's assertion was proved in June 1993 by the British mathematician Andrew Wiles.

                                          

 

The second stamp was released after it was proved by Andrew Wiles.

In 1656 Fermat had started a correspondence with Huygens. This grew out of Huygens interest in probability and the correspondence was soon manipulated by Fermat onto topics of number theory. This topic did not interest Huygens but Fermat tried hard and in New Account of Discoveries in the Science of Numbers sent to Huygens via Carcavi in 1659, he revealed more of his methods than he had done to others. Fermat described his method of infinite descent and gave an example on how it could be used to prove that every prime of the form 4k + 1 could be written as the sum of two squares. 

He died at Castres, Tarn on January 12, 1655. The oldest and most prestigious high school in Toulouse is named after him: 

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Galois

Evariste Galois

Evariste Galois was a French mathematician who made significant contribution to the theory of functions, the theory of equations and number theory. His work became the basis for Galois Theory and Group theory. He was the first to use the word Group as a technical term in mathematics to represent a group of permutations. 

 The mathematical genius Evariste Galois was born on 25^th October 1811 in France. He was the second son of Nicholas Gabriel Galois and Adelaide Marie. His father was the director of a boarding school and later mayor of Bourg – la- Reive. Evariste had a happy childhood. He seemed to have a phenomenal memory.  While still in teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long standing problem. Up to the age of 12, he was educated by his mother who instilled in him knowledge of classics and a sceptical attitude towards religion.

He entered the College Royale de Louis le Grand in Paris in 1823 where his precocious mathematical genius first emerged. He studied the Legendre’s text book Elements de Geometrie and Lagrange’s work on Differential Equation and Analytic Function. By the age of 16, he had published many papers. He appeared for the examination of Ecole Polytechnic but failed. The next year Galois had the good fortune to be studying maths under a distinguished teacher Louis Paul Emile Richard who recognised his exceptional gifts. Richard was enthusiastic about Galois mathematical work and this led to his publication of first paper on continued fraction. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences but the Academy refused to accept them for publication.

On 28 July 1829 Galois's father committed suicide after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt to enter the Polytechnique, and failed yet again. Having been denied admission to the Polytechnique, Galois took the Baccalaureate examinations in order to enter the École Normale. He passed, receiving his degree on 29 December 1829. His examiner in mathematics reported, "This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research."

Galois lived during a time of political turmoil in France. Charles X had succeeded Louis XVIII in 1824, but in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority and Louis Philippe became the king. His anti monarchist view led him to imprisonment leading his expulsion from the college.

Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard. He divided his time between his mathematical work and his political affiliations. Due to controversy surrounding the unit, soon after Galois became a member, on 31 December 1830, the artillery of the National Guard was disbanded out of fear that they might destabilize the government. At around the same time, nineteen officers of Galois' former unit were arrested and charged with conspiracy to overthrow the government. They were later acquitted of all charges.

During his imprisonment, he continued developing his mathematical ideas. On the advice of Simon Poisson he submitted his work on the Theory of Equation for publication but Poisson decleared his work incomprehensible. Galois reacted violently to the rejection and decided to abandon publishing his papers through the Academy and instead publish them privately through his friend Auguste Chevalier. 

During 1830s Galois published several papers in Bulletin des Sciences mathematiques of Baren de Ferussace, a review that normally only published work by established scientist. These articles contained most of his work on Theory of Equation which is now known as Galois Theory.

In 1831, he was arrested again for keeping illegal weapon. He was released on April 29, 1832. On May 25 he wrote to Chevalier, a close friend of his from the Ecole Normale, expressing his complete disenchantment with life, and hinting that a broken love affair was the reason. The woman in question was Stephanie Dumotel, the daughter of the resident physician at the hostel where Galois stayed during last months of his life.

On May 30, Galois fought a duel with pistol and was shot in the abdomen. Galois remained unattended for hours until a passerby took him to the hospital. He refused the services of a priest and died of peritonitis the following day, at the age of 20. A night before the duel was scheduled he recorded his mathematical idea in a later to his former school master Auguste Chevalier where he had out lined his work on elliptical integrals and permutations of groups.

Hermann Weyl, a mathematician, said of this testament, "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind."

Galois' mathematical contributions were published in full in 1843 when Liouville reviewed his manuscript and declared it sound. It was finally published in the October–November 1846 issue of the Journal de Mathématiques Pures et Appliquées. The most famous contribution of this manuscript was a novel proof that there is no quintic formula, that is, that fifth and higher degree equations are not solvable by radicals.

 Unsurprisingly, Galois' collected works amount to only some 60 pages, but within them are many important ideas that have had far-reaching consequences for nearly all branches of mathematics. His work has been compared to that of Niels Abel, another mathematician who died at a very young age, and much of their work had significant overlap.

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Euclid

  Euclid

Euclid of Alexandria is better known as the Father of Geometry. He is one of the most prominent mathematicians of antiquity best known for his treatise The Elements. Little is known about Euclid's life, as there are only a handful of references to him. The date and place of Euclid's birth and the date and circumstances of his death are unknown. But whatever speculated data are available shows that he was born about 325 BC in Alexandria and died about 265 BC.

An Arabian author, al-Qifti (d. 1248), recorded that Euclid's father was Naucrates and his grandfather was Zenarchus, that he was a Greek, born in Tyre and lived in Damascus. But there is no real proof that this is the same Euclid. During the reign of Ptolemy I he taught at Alexandria. Ptolemy had created the great library at Alexandria, which was known as the Museum, because it was considered a house of the muses for the arts and sciences. Many scholars worked and taught there, and that is where Euclid wrote The Elements.

There is a very interesting story about Euclid---

A student who had begun to learn geometry with Euclid asked him “What shall I get by learning all these things in Geometry?” Euclid called his slave and told him to give three pence since he must make gain out of what he learns.

Euclid said to his students that --- There is no royal road to Geometry.

             A page from Euclid’s Elements

Euclid's most famous for his treatise on mathematics called The Elements. The book is a compilation of knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were first proved by Euclid. The Elements is divided into thirteen books which cover plane geometry, arithmetic and number theory, irrational numbers, and solid geometry.

Euclid organized the known geometrical ideas, starting with simple definitions, axioms; formed statements called theorems, and set forth methods for logical proofs. He began with accepted mathematical truths, axioms and postulates, and demonstrated logically 467 propositions in plane and solid geometry. One of the proofs was for the theorem of Pythagoras, proving that the equation is always true for every right triangle. The Elements was the most widely used textbook of all time. It has appeared in more than 1,000 editions since it was first printed in 1482 and is thought to have sold more copies than any book other than the Bible. Original of Euclid's Elements have not been preserved, but Arabic mathematicians obviously had a full copy as an Arabic version of The Elements appeared at the end of the 8th century BC.

 Euclid believed that we can't be sure of any axioms without proof, so he devised logical steps to prove them. There are 5 axioms and 5 postulates found in the book of Euclid. He called Axiom "Common Notions," because they were common to all sciences.

Axioms

1.    Things which are equal to the same thing are also equal to one another.

2.    If equals are added to equals, the sums are equal.

3.    If equals are subtracted from equals, the remainders are equal.

4.    Things which coincide with one another are equal to one another.

5.    The whole is greater than the part.

Postulates

1.      You can draw a straight line between any two points.

2.      You can extend the line indefinitely.

3.      You can draw a circle using any line segment as the radius and one end point as the center.

4.      All right angles are equal.

5.      Given a line and a point, you can draw only one line through the point that is parallel to the first line.

The fifth postulates later became the cause of discovery of new branch of geometry known as Non Euclidian Geometry.

Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. This wonderful book, with all its imperfections, which are indeed slight enough when account is taken of the date it appeared, is and will doubtless remain the greatest mathematical textbook of all time.

Euclid proved that it is impossible to find the "largest prime number," because if you take the largest known prime number, add 1 to the product of all the primes up to and including it; you will get another prime number. Euclid's proof for this theorem is generally accepted as one of the "classic" proofs because of its conciseness and clarity. Millions of prime numbers are known to exist, and more are being added by mathematicians and computer scientists. Mathematicians since Euclid have attempted without success to find a pattern to the sequence of prime numbers.

Although best known for its geometric results, the Elements also include number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

Euclid also wrote Data, which contains 94 propositions, Phaenomena, concerning spherical astronomy, Caloptrics, about mirrors, Optics, the theory of perspective, and a work of music theory. In his works about optics, Euclid made light rays part of geometry, working with them as if they were straight lines. Many of the works ascribed to Euclid are no longer in existence or are incomplete.

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Emmy Noether

Emmy Noether

Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Albert Einstein described her as the most important woman in mathematical history. She lived and worked in Germany during the same period that Ramanujan worked in England and India, but whereas Ramanujan’s contributions were in analysis and number theory, hers were in abstract algebra and in the application of algebra to theoretical physics. She revolutionized the theories of rings, fields and algebras. In physics Noether’s theorem explains the fundamental connection between symmetry and conservation laws. The innovative approach to modern abstract algebra of Emmy Noether not only produced major new results, but also inspired highly productive work by students and colleagues who emulated her techniques.

Amalie Emmy Noether to give her the full name was born on March 23, 1882 in Erlangen in Germany. Her father Max Noether was professor of mathematics at the University of Erlangen while her mother Ida Amalia was the daughter of a wealthy Jewish family of Cologne. She was better known as Emmy. She loved to dance and enjoyed music. She attended the Municipal school for the Higher Education of Daughters until she was 18. Emmy Noether showed early proficiency in French and English. In the spring of 1900 she took the examination for teachers of these languages and received an overall score of very good. Her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the University of Erlangen. This was an unconventional decision because as late as 1900, women were not allowed to enroll in Universities in Germany. Professors frequently refused permission for women event to attend their lectures, and only very rarely women allowed taking university examination. The obstacle in a way could not deter her to enroll in a university, she had to get permission of the professors to take an entrance exam -- she did and she passed, after sitting in on mathematics lectures at the University of Erlangen. She was then allowed to audit courses -- first at the University of Erlangen and then the University of Göttingen, neither of which would permit a woman to attend classes for credit. Finally, in 1904, the University of Erlangen decided to permit women to enroll as regular students, and Emmy Noether returned there. She began to focus solely on mathematics. Under the supervision of Paul Gordan she wrote her dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms). Her dissertation in algebraic math earned her a doctorate summa cum laude in 1908.

Having obtained her doctorate, Noether was well qualified for a position at a university, but the persistent sexiest atmosphere in Germany prevented the brilliant young woman from being able to even apply for a job. This depressed her greatly, but she began helping her father in his research. She also started publishing papers on her own, which were so well received that she was invited to join a number of European Mathematical Societies, including the German Mathematical Association, which had been founded by George Cantor, but still she could not obtain a paying position at a University in Germany. By 1915 Noether was a famous mathematician in her own right and her papers were read with interest throughout the world. They dealt primarily with algebra.

In 1915, Emmy Noether's mentors, Felix Klein and David Hilbert, invited her to join them at the Mathematical Institute in Gottingen, again without compensation. There, she pursued important mathematical work that confirmed key parts of the general theory of relativity. Noether arrived at Gottingen and began her work on invariance in mathematical physics. Meanwhile, Klein rallied to get her appointed a professor at Gottingen, but he had to struggle with the administration until 1919, when his request was finally granted. She became a privatdozent allowing her to teach students and students would pay her directly. In 1922, the University gave her a position as an adjunct professor with a small salary and no tenure or benefits. Soon after arriving at Göttingen, however, she demonstrated her capabilities by proving the theorem now known as Noether's theorem, which shows that a conservation law is associated with any differentiable symmetry of a physical system. 

American physicists Leon M. Lederman and Christopher T. Hill argue in their book Symmetry and the Beautiful Universe that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean Theorem". Her work went far beyond mathematical physics. She made important contributions to Galois Theory, to many other areas of abstract algebra and to topology. Noether was, in fact the greatest algebraist of her time.

Noether’s groundbreaking work in algebra began in 1920 with a paper on non commutative fields. Her work earned her enough recognition that she was invited as a visiting professor in 1928-1929 at the University of Moscow and in 1930 at the University of Frankfurt. In 1932 Emmy Noether and Emil Artin received the Ackermann Teubner Memorial Award for their contributions to mathematics.  Though she was never able to gain a regular faculty position at Göttingen, she was one of many Jewish faculty members who were purged by the Nazis in 1933.  

In America, the Emergency Committee to Aid Displaced German Scholars obtained for Emmy Noether an offer of a professorship at Bryn Mawr College in America, and they paid, with the Rockefeller Foundation, her first year's salary. The grant was renewed for two more years in 1934. This was the first time that Emmy Noether was paid a full professor's salary and accepted as a full faculty member. In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of Abraham Flexner and Oswald Veblen. She also worked with and supervised Abraham Albert and Harry Vandiver. Her time in the United States was pleasant, surrounded as she was by supportive colleagues and absorbed in her favorite subjects. 

In April 1935 doctors discovered a tumor in Noether's pelvis. She was admitted to hospital for surgery to remove a uterine tumor. Although the operation was successful but on April 14, she fell unconscious and she developed a high fever resulting to her death.

Weyl said to her funeral – The memory of her work in science and of her personality among her fellows will not soon pass away. She was a great mathematician, the greatest, I firmly believe, her sex has ever produced, and a great woman.